You want to show that when S is a subspace, we have that span S is contained in S. We don't really need to know anything about the dimension of S.

Take an element y in span S. Since span S consists of finite linear combinations of elements from S, we can write y = a_1x_1+...+a_nx_n for some elements x_i in S, and some scalars a_i (note: this n need not coincide with the dimension of S). And then, as you say, since S is closed under scaling and under vector addition, then a_1x_1+...+a_nx_n will also be in S, whereas y will be in S. So all elements of span S are contained in S - hence span S is a subset of S.